We now recall several different definitions of undirected distance-hereditary graphs [1]}, [2]}.
We therefore need to define the terms of pendant vertices and twins. Let \(G\)  be an undirected, connected graph. A vertex \(v \in V(G)\)  is called pendant if there is \(u \in V(G)\)  such that \(\lbrace u,v\rbrace  \in E(G)\)  and for all other \(w \in V(G)\) , \(w \ne u\)  it holds that \(\lbrace w,v\rbrace  \notin E(G)\) .
A vertex \(v \in V(G)\)  is called twin of \(u \in V(G)\) , if \(N(u) \setminus \lbrace v\rbrace  = N(v) \setminus \lbrace u\rbrace \) .
It is called true twin if \(\lbrace u,v\rbrace  \in E(G)\) , otherwise false twin.
